Statistical mechanics is one of the pillars of modernphysics. It describes howmacroscopic observations (such astemperature andpressure) are related tomicroscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such asheat capacity) to microscopic behavior, whereas, inclassical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.[5]
Boltzmann was born in Erdberg, a suburb ofVienna into aCatholic family. His father, Ludwig Georg Boltzmann, was a revenue official. His grandfather, who had moved to Vienna from Berlin, was a clock manufacturer, and Boltzmann's mother, Katharina Pauernfeind, was originally fromSalzburg. Boltzmann was home-schooled until the age of ten,[6] and then attended high school inLinz,Upper Austria. When Boltzmann was 15, his father died.[7]
Ludwig Boltzmann and co-workers in Graz, 1887: (standing, from the left)Nernst,Streintz,Arrhenius, Hiecke, (sitting, from the left) Aulinger,Ettingshausen, Boltzmann,Klemenčič, Hausmanninger
In 1872, long before women were admitted to Austrian universities, he met Henriette von Aigentler, an aspiring teacher of mathematics and physics in Graz. Her nieces were Slovenian paintersAvgusta Šantel andHenrika Šantel. Henriette was refused permission to audit lectures unofficially. Boltzmann supported her decision to appeal, which was successful. On 17 July 1876 Ludwig Boltzmann married Henriette; they had three daughters: Henriette (1880), Ida (1884) and Else (1891); and a son, Arthur Ludwig (1881).[9] Boltzmann went back toGraz to take up the chair of Experimental Physics. Among his students in Graz wereSvante Arrhenius andWalther Nernst.[10][11] He spent 14 happy years in Graz and it was there that he developed his statistical concept of nature.
Boltzmann was appointed to the Chair of Theoretical Physics at theUniversity of Munich inBavaria, Germany in 1890.
In 1894, Boltzmann succeeded his teacherJoseph Stefan as Professor of Theoretical Physics at the University of Vienna.[12]
Boltzmann spent a great deal of effort in his final years defending his theories.[13] He did not get along with some of his colleagues in Vienna, particularlyErnst Mach, who became a professor of philosophy and history of sciences in 1895. That same yearGeorg Helm andWilhelm Ostwald presented their position onenergetics at a meeting inLübeck. They saw energy, and not matter, as the chief component of the universe. Boltzmann's position carried the day among other physicists who supported his atomic theories in the debate.[14] In 1900, Boltzmann went to theUniversity of Leipzig, on the invitation ofWilhelm Ostwald. Ostwald offered Boltzmann the professorial chair in physics, which became vacant whenGustav Heinrich Wiedemann died. After Mach retired due to bad health, Boltzmann returned to Vienna in 1902.[13] In 1903, Boltzmann, together withGustav von Escherich andEmil Müller, founded theAustrian Mathematical Society. His students includedKarl Přibram,Paul Ehrenfest andLise Meitner.[13]
In Vienna, Boltzmann taught physics and also lectured on philosophy. Boltzmann's lectures onnatural philosophy were very popular and received considerable attention. His first lecture was an enormous success: people stood all the way down the staircase outside the largest available lecture hall, and the Emperor invited him to a reception[when?].[15]
In 1905, he gave an invited course of lectures in the summer session at theUniversity of California in Berkeley, which he described in a popular essayA German professor's trip to El Dorado.[16]
In May 1906, Boltzmann's deteriorating mental condition (described in a letter by the Dean as "a serious form ofneurasthenia") forced him to resign his position. His symptoms indicate he experienced what might today be diagnosed asbipolar disorder.[13][17] Four months later he died by suicide on 5 September 1906, byhanging himself while on vacation with his wife and daughter inDuino, nearTrieste (then Austria).[18][19][20][17]He is buried in the VienneseZentralfriedhof. His tombstone bears the inscription ofBoltzmann's entropy formula:.[13]
Boltzmann'skinetic theory of gases seemed to presuppose the reality ofatoms andmolecules, but almost allGerman philosophers and many scientists likeErnst Mach and thephysical chemistWilhelm Ostwald disbelieved their existence.[21] Boltzmann had been exposed to molecular theory byJames Clerk Maxwell’s paper, "Illustrations of the Dynamical Theory of Gases," which described temperature as dependent on the speed of the molecules. This inspired Boltzmann to embrace atomism, introducing statistics into physics and extending the theory.[22]
Boltzmann wrote treatises on philosophy such as "On the question of the objective existence of processes in inanimate nature" (1897). He was arealist.[23] In his work "On Thesis of Schopenhauer's", Boltzmann refers to his philosophy asmaterialism and says further: "Idealism asserts that only the ego exists, the various ideas, and seeks to explain matter from them. Materialism starts from the existence of matter and seeks to explain sensations from it."[24]
Boltzmann's most important scientific contributions were in thekinetic theory of gases based upon thesecond law of thermodynamics. This was important because Newtonian mechanics did not differentiate between past and futuremotion, butRudolf Clausius’ invention of entropy to describe the second law was based ondisgregation or dispersion at the molecular level so that the future was one-directional. Boltzmann was twenty-five years of age when he came uponJames Clerk Maxwell's work on the kinetic theory of gases which hypothesized thattemperature was caused by collision of molecules. Maxwell used statistics to create a curve of molecular kinetic energy distribution from which Boltzmann clarified and developed the ideas of kinetic theory and entropy based upon statistical atomic theory, creating theMaxwell–Boltzmann distribution as a description of molecular speeds in a gas.[25] It was Boltzmann who derived the first equation to model the dynamic evolution of the probability distribution Maxwell and he had created.[26] Boltzmann's key insight was that dispersion occurred due to the statistical probability of increased molecular "states". Boltzmann went beyond Maxwell by applying his distribution equation to not solely gases, but also liquids and solids. Boltzmann also extended his theory in his 1877 paper beyond Carnot,Rudolf Clausius,James Clerk Maxwell andLord Kelvin by demonstrating that entropy is contributed to by heat, spatial separation, and radiation.[27]Maxwell–Boltzmann statistics and theBoltzmann distribution remain central in the foundations ofclassical statistical mechanics. They are also applicable to otherphenomena that do not requirequantum statistics and provide insight into the meaning oftemperature.
He made multiple attempts to explain the second law of thermodynamics, with the attempts ranging over many areas. He triedHelmholtz's monocycle model,[28][29] a pure ensemble approach like Gibbs, a pure mechanical approach like ergodic theory, the combinatorial argument, theStoßzahlansatz, etc.[30]
Most chemists, since the discoveries ofJohn Dalton in 1808, andJames Clerk Maxwell in Scotland andJosiah Willard Gibbs in the United States, shared Boltzmann's belief inatoms andmolecules, but much of thephysics establishment did not share this belief until decades later. Boltzmann had a long-running dispute with the editor of the preeminent German physics journal of his day, who refused to let Boltzmann refer to atoms and molecules as anything other than convenienttheoretical constructs. Only a couple of years after Boltzmann's death,Perrin's studies ofcolloidal suspensions (1908–1909), based onEinstein's theoretical studies of 1905, confirmed the values of theAvogadro constant and theBoltzmann constant, convincing the world that the tiny particlesreally exist.
To quotePlanck, "Thelogarithmic connection betweenentropy andprobability was first stated by L. Boltzmann in hiskinetic theory of gases".[31] This famous formula forentropyS is[32]wherekB is theBoltzmann constant, and ln is thenatural logarithm.W (forWahrscheinlichkeit, a German word meaning "probability") is the probability of occurrence of amacrostate[33] or, more precisely, the number of possiblemicrostates corresponding to the macroscopic state of a system – the number of (unobservable) "ways" in the (observable)thermodynamic state of a system that can be realized by assigning differentpositions andmomenta to the various molecules. Boltzmann'sparadigm was anideal gas ofNidentical particles, of whichNi are in theith microscopic condition (range) of position and momentum.W can be counted using the formula forpermutationswherei ranges over all possible molecular conditions, and where denotesfactorial. The "correction" in the denominator account forindistinguishable particles in the same condition.
Boltzmann could also be considered one of the forerunners of quantum mechanics due to his suggestion in 1877 that the energy levels of a physical system could be discrete, although Boltzmann used this as a mathematical device with no physical meaning.[34]
An alternative to Boltzmann's formula for entropy, above, is theinformation entropy definition introduced in 1948 byClaude Shannon.[35] Shannon's definition was intended for use in communication theory but is applicable in all areas. It reduces to Boltzmann's expression when all the probabilities are equal, but can, of course, be used when they are not. Its virtue is that it yields immediate results without resorting tofactorials orStirling's approximation. Similar formulas are found, however, as far back as the work of Boltzmann, and explicitly inGibbs (see reference).
Boltzmann's bust in the courtyard arcade of the main building, University of Vienna
The Boltzmann equation was developed to describe the dynamics of an ideal gas.whereƒ represents the distribution function of single-particle position and momentum at a given time (see theMaxwell–Boltzmann distribution),F is a force,m is the mass of a particle,t is the time andv is an average velocity of particles.
This equation describes thetemporal andspatial variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particlephase space. (SeeHamiltonian mechanics.) The first term on the left-hand side represents the explicit time variation of the distribution function, while the second term gives the spatial variation, and the third term describes the effect of any force acting on the particles. The right-hand side of the equation represents the effect of collisions.
In principle, the above equation completely describes the dynamics of an ensemble of gas particles, given appropriateboundary conditions. This first-orderdifferential equation has a deceptively simple appearance, sincef can represent an arbitrary single-particledistribution function. Also, theforce acting on the particles depends directly on the velocity distribution function f. The Boltzmann equation is notoriously difficult tointegrate.David Hilbert spent years trying to solve it without any real success.
The form of the collision term assumed by Boltzmann was approximate. However, for an ideal gas the standardChapman–Enskog solution of the Boltzmann equation is highly accurate. It is expected to lead to incorrect results for an ideal gas only undershock wave conditions.
Boltzmann tried for many years to "prove" thesecond law of thermodynamics using his gas-dynamical equation – his famousH-theorem. However the key assumption he made in formulating the collision term was "molecular chaos", an assumption which breakstime-reversal symmetry as is necessary foranything which could imply the second law. It was from the probabilistic assumption alone that Boltzmann's apparent success emanated, so his long dispute withLoschmidt and others overLoschmidt's paradox ultimately ended in his failure.
Boltzmann's grave in theZentralfriedhof, Vienna, with bust and entropy formula
The idea that thesecond law of thermodynamics or "entropy law" is a law of disorder (or that dynamically ordered states are "infinitely improbable") is due to Boltzmann's view of the second law of thermodynamics.
In particular, it was Boltzmann's attempt to reduce it to astochastic collision function, or law of probability following from the random collisions of mechanical particles. Following Maxwell,[36] Boltzmann modeled gas molecules as colliding billiard balls in a box, noting that with each collision nonequilibrium velocity distributions (groups of molecules moving at the same speed and in the same direction) would become increasingly disordered leading to a final state of macroscopic uniformity and maximum microscopic disorder or the state of maximum entropy (where the macroscopic uniformity corresponds to the obliteration of all field potentials or gradients).[37] The second law, he argued, was thus simply the result of the fact that in a world of mechanically colliding particles disordered states are the most probable. Because there are so many more possible disordered states than ordered ones, a system will almost always be found either in the state of maximum disorder – the macrostate with the greatest number of accessible microstates such as a gas in a box at equilibrium – or moving towards it. A dynamically ordered state, one with molecules moving "at the same speed and in the same direction", Boltzmann concluded, is thus "the most improbable case conceivable...an infinitely improbable configuration of energy."[38]
Boltzmann accomplished the feat of showing that the second law of thermodynamics is only a statistical fact. The gradual disordering of energy is analogous to the disordering of an initially orderedpack of cards under repeated shuffling, and just as the cards will finally return to their original order if shuffled a gigantic number of times, so the entire universe must some-day regain, by pure chance, the state from which it first set out. (This optimistic coda to the idea of the dying universe becomes somewhat muted when one attempts to estimate the timeline which will probably elapse before it spontaneously occurs.)[39] The tendency for entropy increase seems to cause difficulty to beginners in thermodynamics, but is easy to understand from the standpoint of the theory of probability. Consider two ordinarydice, with both sixes face up. After the dice are shaken, the chance of finding these two sixes face up is small (1 in 36); thus one can say that the random motion (the agitation) of the dice, like the chaotic collisions of molecules because of thermal energy, causes the less probable state to change to one that is more probable. With millions of dice, like the millions of atoms involved in thermodynamic calculations, the probability of their all being sixes becomes so vanishingly small that the systemmust move to one of the more probable states.[40]
Ludwig Boltzmann's contributions to physics and philosophy have left a lasting impact on modern science. His pioneering work in statistical mechanics and thermodynamics laid the foundation for some of the most fundamental concepts in physics. For instance,Max Planck in quantizing resonators in hisBlack Body theory of radiation used theBoltzmann constant to describe the entropy of the system to arrive at his formula in 1900.[41] However, Boltzmann's work was not always readily accepted during his lifetime, and he faced opposition from some of his contemporaries, particularly in regard to the existence of atoms and molecules. Nevertheless, the validity and importance of his ideas were eventually recognized, and they have since become cornerstones of modern physics. Here, we delve into some aspects of Boltzmann's legacy and his influence on various areas of science.
Atomic theory and the existence of atoms and molecules
Boltzmann's kinetic theory of gases was one of the first attempts to explain macroscopic properties, such as pressure and temperature, in terms of the behaviour of individual atoms and molecules. Although many chemists were already accepting the existence of atoms and molecules, the broader physics community took some time to embrace this view. Boltzmann's long-running dispute with the editor of a prominent German physics journal over the acceptance of atoms and molecules underscores the initial resistance to this idea.
It was only after experiments (includingJean Perrin's studies of colloidal suspensions) confirmed the values of the Avogadro constant and the Boltzmann constant that the existence of atoms and molecules gained wider acceptance. Boltzmann's kinetic theory played a crucial role in demonstrating the reality of atoms and molecules and explaining various phenomena in gases, liquids, and solids.
Statistical mechanics, which Boltzmann pioneered, connects macroscopic observations with microscopic behaviors. His statistical explanation of the second law of thermodynamics was a significant achievement, and he provided the current definition of entropy (), wherekB is the Boltzmann constant and Ω is the number of microstates corresponding to a given macrostate.
Max Planck later named the constantkB as the Boltzmann constant in honor of Boltzmann's contributions to statistical mechanics. The Boltzmann constant is now a fundamental constant in physics and across many scientific disciplines.
Because theBoltzmann equation is practical in solving problems in rarefied or dilute gases, it has been used in many diverse areas of technology. It has been used to calculateSpace Shuttle re-entry in the upper atmosphere.[42] It is the basis forneutron transport theory, and ion transport insemiconductors.[43][44]
Boltzmann's work in statistical mechanics laid the groundwork for understanding the statistical behavior of particles in systems with a large number of degrees of freedom. His 1877 paper on the kinetic theory of heat used discrete energy levels of physical systems as a mathematical device, and went on to show that the same approach could be applied to continuous systems. This might be seen as a forerunner to the development of quantum mechanics.[45] One biographer of Boltzmann says that Boltzmann’s approach “pav[ed] the way for Planck.”[46]
Quantization of energy levels became a fundamental postulate in quantum mechanics, leading to groundbreaking theories likequantum electrodynamics andquantum field theory. Thus, Boltzmann's early insights into the quantization of energy levels had a profound influence on the development of quantum physics.
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^Partington, J.R. (1949),An Advanced Treatise on Physical Chemistry, vol. 1,Fundamental Principles,The Properties of Gases, London:Longmans, Green and Co., p. 300
^Jäger, Gustav; Nabl, Josef; Meyer, Stephan (April 1999). "Three Assistants on Boltzmann".Synthese.119 (1–2):69–84.doi:10.1023/A:1005239104047.S2CID30499879.Paul Ehrenfest (1880–1933) along with Nernst, Arrhenius, and Meitner must be considered among Boltzmann's most outstanding students.
^Muir, Hazel,Eureka! Science's greatest thinkers and their key breakthroughs, p.152,ISBN1-78087-325-5
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^Upon Boltzmann's death,Friedrich ("Fritz") Hasenöhrl became his successor in the professorial chair of physics at Vienna.
^Cercignani, Carlo (2008).Ludwig Boltzmann: the man who trusted atoms (Repr ed.). Oxford: Oxford Univ. Press. p. 176.ISBN978-0-19-850154-1.
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^Maxwell, J. (1871). Theory of heat. London: Longmans, Green & Co.
^Boltzmann, L. (1974). The second law of thermodynamics. Populare Schriften, Essay 3, address to a formal meeting of the Imperial Academy of Science, 29 May 1886, reprinted in Ludwig Boltzmann, Theoretical physics and philosophical problem, S. G. Brush (Trans.). Boston: Reidel. (Original work published 1886)
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^Neunzert, H., Gropengießer, F., Struckmeier, J. (1991). Computational Methods for the Boltzmann Equation. In: Spigler, R. (eds) Applied and Industrial Mathematics. Mathematics and Its Applications, vol 56. Springer, Dordrecht.doi:10.1007/978-94-009-1908-2_10
^AN OVERVIEW OF THE BOLTZMANN TRANSPORT EQUATION SOLUTION FOR NEUTRONS, PHOTONS AND ELECTRONS IN CARTESIAN GEOMETRY, Ba ́rbara D. do Amaral Rodriguez, Marco Tu ́llio Vilhena, 2009 International Nuclear Atlantic Conference - INAC 2009 Rio de Janeiro, RJ, Brazil, September 27 to October 2, 2009 ASSOCIAC ̧A ̃OBRASILEIRADEENERGIANUCLEAR-ABENISBN978-85-99141-03-8
^Sharp, K.; Matschinsky, F. Translation of Ludwig Boltzmann’s Paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI 1877, pp 373-435 (Wien. Ber. 1877, 76:373-435). Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, p. 164-223, Barth, Leipzig, 1909. Entropy 2015, 17, 1971-2009.https://doi.org/10.3390/e17041971https://www.mdpi.com/1099-4300/17/4/1971
^Carlo Cercignani, “Ludwig Boltzmann: The Man Who Trusted Atoms,” Chap. 12.3 Black-Body Radiation, 2006,ISBN978-0198570646.
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